Brick-wall lattice paths and applications (original) (raw)

The Shape of the Reliability Polynomial of a Hammock Network

Advances in Intelligent Systems and Computing, 2020

Motivated by the study of hammock (aka brick-wall) networks, we introduce in this paper the notion of X-path. Using the Jordan Curve Theorem for piecewise smooth curves, we prove duality properties for hammock networks. Consequences for reliability polynomials are given.

Explicit Formulas for Enumeration of Lattice Paths: Basketball and the Kernel Method

Lattice Path Combinatorics and Applications, 2019

This article deals with the enumeration of directed lattice walks on the integers with any finite set of steps, starting at a given altitude j and ending at a given altitude k, with additional constraints such as, for example, to never attain altitude 0 in-between. We first discuss the case of walks on the integers with steps −h,. .. , −1, +1,. .. , +h. The case h = 1 is equivalent to the classical Dyck paths, for which many ways of getting explicit formulas involving Catalan-like numbers are known. The case h = 2 corresponds to "basketball" walks, which we treat in full detail. Then we move on to the more general case of walks with any finite set of steps, also allowing some weights/probabilities associated with each step. We show how a method of wide applicability, the so-called "kernel method", leads to explicit formulas for the number of walks of length n, for any h, in terms of nested sums of binomials. We finally relate some special cases to other combinatorial problems, or to problems arising in queuing theory.

On a Reliability Polynomial of Fibonacci and Some Other Graphs

2003

Abstract:-This paper investigates the reliability polynomial of a Fibonacci graph. Trying to find an efficient way of computing the coefficients of this polynomial, we show that for 1≤ k≤ 3 the absolute value of its k-th coefficient is equal to the number of k-edge mincuts in this Fibonacci graph. Further, we generalize this result to some other directed acyclic graphs. Key-Words:-coefficients of reliability polynomial, Fibonacci graph, mincut, operating path, probabilistic graph, reliability, reliability polynomial, size of mincut, st-connectedness, st-dag

Refinements of Lattice paths with flaws

The classical Chung-Feller theorem [2] tells us that the number of Dyck paths of length nnn with mmm flaws is the nnn-th Catalan number and independent on mmm. In this paper, we consider the refinements of Dyck paths with flaws by four parameters, namely peak, valley, double descent and double ascent. Let pn,m,k{p}_{n,m,k}pn,m,k be the number of all the Dyck paths of semi-length nnn with mmm flaws and kkk peaks. First, we derive the reciprocity theorem for the polynomial Pn,m(x)=sumlimitsk=1npn,m,kxkP_{n,m}(x)=\sum\limits_{k=1}^np_{n,m,k}x^kPn,m(x)=sumlimitsk=1npn,m,kxk. Then we find the Chung-Feller properties for the sum of pn,m,kp_{n,m,k}pn,m,k and pn,m,n−kp_{n,m,n-k}pn,m,n−k. Finally, we provide a Chung-Feller type theorem for Dyck paths of length nnn with kkk double ascents: the number of all the Dyck paths of semi-length nnn with mmm flaws and kkk double ascents is equal to the number of all the Dyck paths that have semi-length nnn, kkk double ascents and never pass below the x-axis, which is counted by the Narayana number. Let vn,m,k{v}_{n,m,k}vn,m,k (resp. dn,m,kd_{n,m,k}dn,m,k) be ...

Bell polynomials and -generalized Dyck paths

Discrete Applied Mathematics, 2008

A k-generalized Dyck path of length n is a lattice path from (0, 0) to (n, 0) in the plane integer lattice Z × Z consisting of horizontal-steps (k, 0) for a given integer k ≥ 0, up-steps (1, 1), and down-steps (1, −1), which never passes below the x-axis. The present paper studies three kinds of statistics on k-generalized Dyck paths: "number of u-segments", "number of internal u-segments" and "number of (u, h)-segments". The Lagrange inversion formula is used to represent the generating function for the number of k-generalized Dyck paths according to the statistics as a sum of the partial Bell polynomials or the potential polynomials. Many important special cases are considered leading to several surprising observations. Moreover, enumeration results related to u-segments and (u, h)-segments are also established, which produce many new combinatorial identities, and specially, two new expressions for Catalan numbers.

Enumeration of generalised directed lattice paths in a strip

2019

This thesis is about the enumeration of two models of directed lattice paths in a strip. The first problem considered is of path diagrams formed by Dyck paths and columns underneath it, counted with respect to the length of the paths and the sum of the heights of the columns. The enumeration of these path diagrams is related to q-deformed tangent and secant numbers. Generating functions of height-restricted path diagrams are given by convergents of continued fractions. We derive expressions for these convergents in terms of basic hypergeometric functions, leading to a hierarchy of novel identities for basic hypergeometric functions. From these expressions, we also find novel expressions for the infinite continued fractions, leading to a different proof of known enumeration formulas for q-tangent and q-secant numbers. The second problem considered is the enumeration of directed weighted paths in a strip with arbitrary step heights. Here, we find an appealing formula for their generat...

Lattice Paths and Faber Polynomials

Advances in Combinatorial Methods and Applications to Probability and Statistics, 1997

The rth Faber polynomial of the Laurent series f (t) = t + f 0 + f 1 /t + f 2 /t 2 + · · · is the unique polynomial F r (u) of degree r in u such that F r (f ) = t r + negative powers of t. We apply Faber polynomials, which were originally used to study univalent functions, to lattice path enumeration.

Circular Digraph Walks, k-Balanced Strings, Lattice Paths and Chebychev Polynomials

The electronic journal of combinatorics

We count the number of walks of length n on a k-node circular digraph that cover all k nodes in two ways. The first way illustrates the transfer-matrix method. The second involves counting various classes of height-restricted lattice paths. We observe that the results also count so-called k-balanced strings of length n, generalizing a 1996 Putnam problem.

A Continuous Analogue of Lattice Path Enumeration

The Electronic Journal of Combinatorics

Following the work of Cano and Díaz, we consider a continuous analog of lattice path enumeration. This process allows us to define a continuous version of many discrete objects that count certain types of lattice paths. As an example of this process, we define continuous versions of binomial and multinomial coefficients, and describe some identities and partial differential equations that they satisfy. Finally, as an important byproduct of these continuous analogs, we illustrate a general method to recover discrete combinatorial quantities from their continuous analogs, via an application of the Khovanski-Puklikov discretizing Todd operators.